A d-dimensional cellular learning automata is a structure A = (Zd,Φ,A,N,F), where
(i) Zd is a lattice of d-tuples of integer numbers.
(ii) Φ is a finite set of states.
(iii) A is the set of LAs each of which is assigned to each cell of the CA.
(iv) N = {¯x1, ¯x2, . . . , ¯x ¯m} is a finite subset of Zd called neighborhood vector, where ¯m represents the number of neighboring cells and ¯xi ∈ Zd. The neighborhood vector determines the relative position of the neighboring cells from any given cell u in the lattice Zd. The neighbors of a particular cell u are a set of cells {u ¯xi|i = 1, 2, . . . , ¯m}. We assume that, there exists a neighborhood function
¯N(u) mapping a cell u to the set of its neighbors, that is
¯N(u) = (u ¯x1, u ¯x2, . . . , u ¯x ¯m).
For the sake of simplicity, we assume that the first element of neighborhood vector (i.e. ¯x1) is equal to d-tuple (0, 0, . . . , 0). The neighborhood function ¯N(u) must satisfy in the two following conditions:
— u ∈ ¯N (u) for all u ∈ Zd.
— u1 ∈ ¯N (u2) ⇔ u2 ∈ ¯N (u1) for all u1, u2 ∈ Zd.
(v) F : Φ¯m → β is the local rule of the cellular learning automata, where β is the set of values that the reinforcement signal can take. It gives the reinforcement signal to each LA from the current actions selected by its neighboring LAs.
A number of applications for CLA have been developed recently such as rumor diffusion, image processing, modeling of commerce networks, fixed channel assignment in cellular networks, and VLSI Placement to mention a few. The CLA can be classified into synchronous and asynchronous. In synchronous CLA, all cells are synchronized with a global clock and executed at the same time. Also a mathematical methodology to study the steady state behavior of the synchronous CLA is given and its convergence properties has been investigated. It is shown that the synchronous CLA converges to a globally stable state for a class of rules called commutative rules.

There is a lab named SOFTLAB that works in this area under supervision of Prof. Mohammad Reza Meybodialso you can refer to Amir Hossein Momeni Azandaryani who is Phd student in this lab.